关于连续介质扩散通量的结构--一种新的封闭格式及其与Maxwell-Stefan和Fick-Onsager方法的关系
On the structure of continuum thermodynamical diffusion fluxes -- A novel closure scheme and its relation to the Maxwell-Stefan and the Fick-Onsager approach
摘要：
本文回顾了不可逆过程热力学框架内的多组分扩散模型。我们简要回顾了两个著名的主要方法，导致广义Fick-Onsager多组分扩散通量或广义Maxwell-Stefan方程。后一种方法的优点是，所得到的通量是一致的非负的非奇异和非退化的麦克斯韦-斯特凡扩散系数的部分质量密度。另一方面，这种方法需要计算昂贵的矩阵求逆，因为通量仅隐含地给出。我们提出并讨论了一种新的和更直接的封闭，避免了反演的麦克斯韦-斯特凡方程。结果表明，在浓度为正的自然要求下，这三种封闭实际上是等价的，从而揭示了连续介质扩散通量的一般结构。作为一个特殊的情况下，新的封闭也产生了一个核心对角扩散模型，其中只有那些交叉效应是必要的，以保证与总质量守恒的一致性，加上组成的扩散系数的依赖性。这种核心对角封闭原来提供了一个严格的基础，最近扩展的Darken方程从二元混合物的一般多组分的情况下。作为我们的调查结果，我们还解决了不同的问题，多组分热力学或Fickian扩散系数的符号。我们严格表明，在一般情况下，第二定律需要积极的张量和运营商的性质，而不是标量扩散。
英文摘要：
This paper revisits the modeling of multicomponent diffusion within the framework of thermodynamics of irreversible processes. We briefly review the two well-known main approaches, leading to the generalized Fick-Onsager multicomponent diffusion fluxes or to the generalized Maxwell-Stefan equations. The latter approach has the advantage that the resulting fluxes are consistent with non-negativity of the partial mass densities for non-singular and non-degenerate Maxwell-Stefan diffusivities. On the other hand, this approach requires computationally expensive matrix inversions since the fluxes are only implicitly given. We propose and discuss a novel and more direct closure which avoids the inversion of the Maxwell-Stefan equations. It is shown that all three closures are actually equivalent under the natural requirement of positivity for the concentrations, thus revealing the general structure of continuum thermodynamical diffusion fluxes. As a special case, the new closure also gives rise to a core-diagonal diffusion model in which only those cross-effects are present that are necessary to guarantee consistency with total mass conservation, plus a compositional dependence of the diffusivity. This core-diagonal closure turns out to provide a rigorous fundament for recent extensions of the Darken equation from binary mixtures to the general multicomponent case. As an outcome of our investigation, we also address different questions related to the sign of multicomponent thermodynamic or Fickian diffusion coefficients. We show rigorously that in general the second law requires positivity properties for tensors and operators rather than for scalar diffusivities.
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